# All Questions

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### Epsilon-Delta Theorem for Darboux Integrability

Trying to prove the epsilon delta theorem for Darboux Integrability using the fact that U(f,P)-U(f,Q)≤2NBmesh(P) and L(f,Q)-L(f,P)≤ 2NBmesh(P) where N is the number of elements contained in Q but not ...
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### Proof of the Gelfand-Naimark Theorem

I am reading a proof of the Gelfand-Naimark theorem in Principles of Harmonic Analysis by Anton Deitmar and Siegfried Echterhoff. I have questions about some of the steps. Theorem. If $A$ is a ...
• 11.4k
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### Validity of a proof of the Fisher Information Data Processing inequality, $I[f(X)] \le I[X]$.

I'm trying to prove that taking a function of a random variable never creates a better estimator (in the terms of Fisher information) than using the original random variable directly. I have a proof (...
• 4,326
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### Kronecker sum as a Kronecker product

I seek the following relationship (if there is one so): 𝐶⊗𝐷=(𝐴⊗I)+(I⊗D) I would like to obtain A=𝑓(C,D) We can assume D, circulant matrices, square and symmetric for simplicity (then all term of ...
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### Colouring a map with disconnected regions

Consider a map of countries, where each country may consist of up to $n$ disconnected regions. Show that the minimum number of colours needed to colour this map is at most $6n$. We require that all ...
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### Recovery Time for Logistic model equation with harvesting

We are given a logistic growth model with constant harvesting as: $\frac{dN}{dt} = rN(1-\frac{N}{K})-Y_0$ We are asked to show that the recovery time for harvesting a yield $Y_0$, $T_R(Y_0)$, ...
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### Let a,b,c be positive reals such that $a^2 + b^2 + c^2 \leq 3$. Prove $\sqrt {1+a} +\sqrt {1+b} +\sqrt {1+c} \geq \frac{\sqrt 2}{3} *(a+b+c)^2$

I am unsure of where to even start. I know that $$\sqrt 2 \geq \frac{\sqrt 2}{3} *(a^2 +b^2 +c^2)$$ which is needed for solving it, but otherwise I am unsure of what to do.
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### Trouble with converting the negation of a formula to CNF

I'm trying to convert the negation of the following formula to CNF: (p → (q → r)) → ((p → q) → (p → r)) These are the steps I am following: ¬((p → (q → r)) → ((p → q) → (p → r))) ¬(¬(¬p ∨ (¬q ∨ r)) ∨ (...
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### Efficient way to compute average of outer products.

I have a ser of vectors $v_i \in \mathbb{R}^d$, with $i \in \{1, ..., n\}$. I have to compute a weighted average of the outer products of these vectors: \begin{equation} \sum_{i=1}^{n} v_i v_i^T w_i \...
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### Upper bound of the dot product of two real-valued vectors

I am interested in proving the following: $\sum_{i=1}^n a_ib_i \leq \max \{\sum_{i=1}^n a^2_i,\sum_{i=1}^n b^2_i\}$, where $a_i,b_i \in \mathbb{R}$. I came up with a geometric proof and a proof by ...
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1 vote
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### Property of Lebesgue Density Point

I have this claim concerning a Lebesgue density point that I need to understand rigorously: Let $0$ be a density point of a closed set $A\subset \mathbb{R}^n$. Then for any $x\in\mathbb{R}^n$ there ...
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### Question on subspace/dimension

Is it possible to find two vector subspaces $V$ and $W$ of $R^3$ with $V \cap W=\{\vec{0}\}$ so that $\dim V=\dim W =2$?
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### Find all solution of $(n-1)\lfloor x \rfloor + n\lceil x \rceil = (n+1)\{x\}$

Given that $n$ is an integer. I have to find all values of $x$ that satisfy the equation.
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### Weierstrass-Erdmann condition in light reflection

I have a calculus of variations problem with discontinuities where the Weierstrass-Erdmann condition seems impossible to satisfy. Hoping someone can figure out what I did wrong. I consider light ...
• 1
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### Analogy of Archimedean Property of powers of rational numbers

For any $a,b \in \mathbb{Q}$, $a > 1, b> 0$, can we find an integer $n$ such that $a^n > b$? (Can we prove this without using any property of real numbers and hence functions like logarithm ...
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19 views

### Evaluation of a Trigonometric Limit...

$$\lim_{x\to\tfrac{\pi}{4}} \frac{(\cos x + \sin x)^3 - 2\sqrt2}{1 - \sin 2x}$$ I tried solving this question without using L'Hospital's Rule but couldn't find a satisfactory way of approach to solve ...
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### Evaluate area bounded by curve S=0, its Latus Rectum and X-axis

Several equilateral triangles with respective sides 1, 2 ,3 ...n (n is a natural number) are placed end to end, starting from origin, one side of each lying on X-axis in the first quadrant. If the ...
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1 vote
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### Doubly transitive action on upper half plane.

Can I transform the triangle in the right hand side to a triangle in the left hand side by a Mobius transform? If $PSL(2, \mathbb{R})$ acts doubly transitive on $\mathbb{R} \cup \infty$, then do we ...
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1 vote
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### Intersection of perpendicular tangent lines - generalization of directrix?

This is a funny little problem that I came up with. For a differentiable function $f$, define a locus of points $P$ as follows: Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
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### I'm playing a game where I'm tossing a fair coin. When it's heads I gain a dollar. When it's tails I lose a dollar. I play until I have +24 or -36

I'm playing a game where I'm tossing a fair coin. When it's heads I gain a dollar. When it's tails I lose a dollar. I play until I have +24 or -36. What's the probability of each condition? I'm ... 38 views

### What is the generalized solution to $M = \int_0^1 (1-x^m)^n$?

What is the generalized solution to $$M = \int_0^1 (1-x^m)^n \,dx$$ (where it can be expressed in terms of n and m) and m and n are rational numbers if considering m and n as natural numbers makes ...
• 250
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### If $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are b and c equal?

Is it true that, if $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are $b$ and $c$ equal? For coprime $(a,b)$ and $(a,c)$, it is trivial. It is also easy when $a|b$ and $a|c$. I am unable to construct ...
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### Christoffel Symbol in terms of determinant of the metric

It is generally known that, if g is the metric, $$Γ^𝜇_{\mu\alpha}= \partial_{\alpha}(ln\sqrt{|g|}).$$ It is also known that Christoffels are symmetric on its second and third components, meaning ...
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### How do we evaluate this REALLY tricky integral?

$$\int{\frac{\cos 9x + \cos 6x}{2 \cos 5x-1} dx}$$ The objective is to find the answer in terms of $\sin 4x$ and $\sin x$ I would like to share my attempt and then ask a conceptual doubt as usual but ...
11 views

### Showing that $\{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$ is a $T$ invariant set

Let $(X, \sigma, \mu)$ be a probability space, $T:X\to X$ a $\mu$-invariant map and $f\in L^1(\mu)$. Define $A := \{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$. I would like to conclude that $A$ ...
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### Markov chains: Prove vector of mean hitting times is minimal non-negative solution

Definitions: Let $H^A = \inf \{n \geq 0 : X_n \in A \}$ be the first hitting-time of the set $A$. Let $k_i^A = \mathbb{E}_i(H^A)$ be the mean hitting time of $A$ from $i$. Theorem: The vector of mean ...
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### Hadamard Gap Theorem and Lacunary Functions

Does anyone know any good reference or even a simple proof for Hadamard Gap Theorem or even just the fact that a lacunary function diverges at 1 (I mean the limit not just the evaluation). In fact, I ...
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### Vector Newton's method without using matrix inverse?

For a set of non liner equations $f_i(\vec x)$ were $i\in \mathbb{N}$ was the index, one can construct a vector \begin{equation} F(\vec x)= \begin{pmatrix} f_1{\vec x} \\ f_2(\vec x) \end{pmatrix} \...
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### Evaluate Integral : $\int\;\frac {1}{\sqrt {1-e^{2x}}}dx$

How to evaluate Integral : $$\int\;\frac {1}{\sqrt {1-e^{2x}}}dx$$ My attemp: I know that my answer is wrong, but I don't know exactly where the error is. I need help finding out? I = \int\;\frac {...
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### Is the following conjecture relating to the chromatic number correct

Define $\mu(G)$ as the minimum number $k$ so that we can partition $V$ in $n, 1<n<k+1$ sets $V_i$ and so that every set $V_i$ has less than $k$ edges between $V_i$ and $V_i^c$ Let $G=(V,E)$ be ...
1 vote
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### The linear system associated to a blow-up(of surfaces)

Take a point $p\in\mathbb{P}^2_k$ and blow it up. Then we get a morphism $f:X \rightarrow\mathbb{P}^2_k$ from the blow-up. This should then be describe by a linear system on $X$. What is that linear ...